nLab
realizability

Idea

The idea of realizability is a way of making the Brouwer-Heyting-Kolmogorov interpretation of constructivism and intuitionistic mathematics precise. It is related to the propositions as types paradigm. For instance, constructively a proof of an existential quantification $\underset{x\in X}{\exists} \phi(x)$ consists of constructing a specific $x \in X$ and a proof of $\phi(x)$, which “realizes” the truth of the statement, whence the name (see e.g. Streicher 07, section 1, Vermeeren 09, section 1 for introductions to the rough idea, or Bauer 05, page 12 and def. 4.7 for an actual definition).

Related entries

• realizability topos

• realizability model

• realizability interpretation

• effective topos, Kleene-Vesley topos

computability

	type I computability	type II computability
typical domain	natural numbers $\mathbb{N}$	Baire space of infinite sequences $\mathbb{B} = \mathbb{N}^{\mathbb{N}}$
computable functions	partial recursive function	computable function (analysis)
type of computable mathematics	recursive mathematics	computable analysis, Type Two Theory of Effectivity
type of realizability	number realizability	function realizability
partial combinatory algebra	Kleene's first partial combinatory algebra	Kleene's second partial combinatory algebra

References

Realizability originates with the interpretation of intuitionistic number theory, later developed as Heyting arithmetic, in

• Stephen Kleene, On the interpretation of intuitionistic number theory Journal of Symbolic Logic, 10:109–124, 1945. link

A historical survey of realizability (including categorical realizability) is in

• Jaap van Oosten, Realizability: An Historical Essay, 2000 (link, pdf)

A quick survey is in

• Stijn Vermeeren, Realizability Toposes, 2009 (pdf)

being a summary of

• Martin Hyland, The effective topos, in The L.E.J. Brouwer Centenary

  Symposium (A. S. Toelstra and D. van Dalen, eds.), North-Holland Publishing Company, 1982, pp. 165–216

A modern textbook account is

• Jaap van Oosten, Realizability: an introduction to its categorical side, Studies in Logic and the Foundations of Mathematics, vol. 152, Elsevier, 2008 (preface pdf)

Lecture notes include

• Thomas Streicher, Realizability (2007/08) (pdf)

and

• Andrej Bauer, Realizability as connection between constructive and computable mathematics, in T. Grubba, P. Hertling, H. Tsuiki, and Klaus Weihrauch, (eds.) CCA 2005 - Second International Conference on Computability and Complexity in Analysis, August 25-29,2005, Kyoto, Japan, ser. Informatik Berichte, , vol. 326-7/2005. FernUniversität Hagen, Germany, 2005, pp. 378–379. (pdf)

based on

• Andrej Bauer, The Realizability Approach to

  Computable Analysis and Topology_, PhD thesis CMU (2000) (pdf)

• Peter Lietz, From Constructive Mathematics to Computable Analysis via the Realizability Interpretation (pdf.gz)

See also

• Steven Awodey, Andrej Bauer, Sheaf toposes for realizability (pdf)

• Wouter Pieter Stekelenburg, Realizability Categories, PhD thesis, Utrecht 2013 (arXiv:1301.2134)

• Martin Hyland, Variations on Realizability (2000) (web)